Simplify the following expression and state the condition under which the simplification is valid: $p = \dfrac{t^2 - t - 2}{t^2 + 9t + 8}$
Answer: First factor the expressions in the numerator and denominator. $ \dfrac{t^2 - t - 2}{t^2 + 9t + 8} = \dfrac{(t - 2)(t + 1)}{(t + 8)(t + 1)} $ Notice that the term $(t + 1)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(t + 1)$ gives: $p = \dfrac{t - 2}{t + 8}$ Since we divided by $(t + 1)$, $t \neq -1$. $p = \dfrac{t - 2}{t + 8}; \space t \neq -1$